Global dynamics conjecture for the competition system with delay

Consider the competition system

$$\dot{x}_1(t)=\cdots,\qquad \dot{x}_2(t)=\cdots$$

with positive initial data, basic reproduction numbers $\mathcal{R}_0^{(1)}>1$ and $\mathcal{R}_0^{(2)}>1$, boundary equilibria $E_1,E_2$, and coexistence equilibrium $E_c$. Let $x_1^*$ and $x_2^*$ denote the corresponding single-species equilibrium values. Global dynamics conjecture. For every solution $(x_1(t),x_2(t))$, the following assertions hold: (a) if $\alpha_1x_1^*>\kappa_2x_2^*$ and $\kappa_1x_1^*>\alpha_2x_2^*$, then $(x_1(t),x_2(t))$ converges to $E_1$; (b) if $\alpha_1x_1^*<\kappa_2x_2^*$ and $\kappa_1x_1^*<\alpha_2x_2^*$, then it converges to $E_2$; (c) if condition $\mathrm{(H_s)}$ holds, then it converges to $E_c$; and (d) if condition $\mathrm{(H_u)}$ holds, then every solution whose initial data is not on the one-dimensional stable manifold of $E_c$ converges to either $E_1$ or $E_2$. The conjecture proposes a complete classification of the global dynamics in the parameter regimes considered, extending the local stability analysis and numerical bifurcation diagrams; the assertions remain to be established analytically.